The Bradford distribution differs from most probability distributions in that it is concerned with the rank-order S of the elements in terms of their productivity (from highest down to lowest) rather than with the numerical values n of the element's productivity. The defining relationship is that S is exponentially related to G, the cumulative production of the elements of rank-order S or less. This implies a Zipf-like relationship between mean productivity and rank-order, which is analogous to the Weber-Fechner law of Psychophysics. A variational specification of the distribution is given, and it is pointed out that the relationship between the construction of the Bradford and that of the usual distributions is roughly analogous to the relationship between Lebesgue and Riemann integration. It has been pointed out in the past that many informational data fit the approximate formula for the Bradford distribution (where n is considered to be a continuous variable). It is shown that when the exact Bradford distribution is used (with productivity taken to be an integer, as it actually is) then the fit with the data is even better, clear down to n=3,2 and even 1. This is demonstrated by fits with data from the scatter of articles on operations research among journals and also with data on the citations to a single medical journal by articles in other journals. The paper also includes tables and formulas to enable the reader to fit the distribution to data of his choice.